Reciprocal Lattice Concept

Diffraction phenomena can be interpreted most conveniently with the aid of the reciprocal lattice concept. A plane can be represented by a line drawn normal to the plane the spatial orientation of this line describes the orientation of the plane. Furthermore, the length of the line can be fixed in an inverse proportion to the interplanar spacing of the plane it represents. 

When a normal is drawn to each plane in a crystal and normals are drawn from a common origin, the terminal points of these normals constitute a lattice array. This is called the reciprocal lattice (Birks, 1953 Bragg, 1933) because of the distance of each point from the origin is reciprocal to the interplanar spacing of the planes that it represents. In an individual cell of a crystalline structure, there exists near the origin the traces of several planes in a unit cell of a crystal, namely, the (100), (001), (101), and (102) planes. Normals to these planes are called the reciprocal lattice vectors ahki and they are defined by 

In three dimensions, the lattice array is described by three reciprocal lattice vectors whose magnitudes are given by 

There directions are defined by three interaxial angles a, p, y.  

Writing the Bragg equation in a form that relates the glancing angle 0 most clearly to the other parameters, we have 

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The usefulness of the reciprocal lattice concept may be once again demonstrated here by illustrating how easily Eqs. 2.29 to 2.34 can be derived in the reciprocal space employing reciprocal lattice vectors. When the derivation is performed in the direct space the geometrical considerations become quite complex. 

The reciprocal lattice concept can be illustrated by constructing a Fourier series in the space of 2D periodic functions. Let /(r) be a periodic function of a radius-vector in the surface plane, i.e.,... 

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

The l index of every spot is thus obvious by mere inspection. The other two indices are best obtained by a graphical method. Just as all spots with the same l indices (in the present example) lie on definite lines, so all spots with the same hk values lie on definite curves. But these (hk curves have a form less simple than the l curves . The form of these curves is most readily determined by introducing a piece of mental scaffolding known as the reciprocal lattice —a conception which has proved to be a tool of the greatest value for the solution of all geometrical problems concerned with the directions of X-ray reflections from crystals. It was introduced by Ewald (1921). 

For monoclime and triclinic cells, the formulae for the spacings are very unwieldy. Graphical methods based on the conception of the reciprocal lattice are recommended (Chapter VI). 

For triclinic crystals the expression is so unwieldy that it is not worth while attempting to use it a graphical method based on the conception of the reciprocal lattice should be used (see pp. 154 ff). The reciprocal lattice method is also more rapid than calculation for monoclinic crystals. 

Note also that in a triclinic crystal a and a are not collinear in a monoclinic crystal (b unique setting) b is parallel to b, but a and c form the obtuse angle [>, while a and c form a smaller acute angle /T given by fJ = 180 — fi. The reciprocal lattice vectors and the direct lattice vectors are a ying-yang duo of concepts, as are position space and momentum space, or space domain and time domain. Fourier transformation helps us walk across from one space to other, as convenience dictates Some problems are easy in one space, others in the space dual to it this amphoterism is frequent in physics. The directions of the direct and reciprocal lattice vectors are shown as face normals in Fig. 7.22. 

These and other calcnlations can be greatly simplified, using the concept of the reciprocal space (which has the dimension of reciprocal length, A ) and of reciprocal crystal lattice in this space. The reciprocal lattice vectors, a, b, and c, are related to those of the direct lattice (a, b, and c) by the relationships... 

In order to extend the description of electronic band structures to general 2D and 3D systems, it is convenient to introduce the concepts of the direct and reciprocal lattices. For a general 3D lattice with repeat vectors a, b, and c, the positions of lattice sites ma, nb, pc) can be represented as vectors R... 

In the Bragg formulation of diffraction we thus refer to reflections from lattice planes and can ignore the positions of the atoms. The Laue formulation of diffraction, on the other hand, considers only diffraction from atoms but can be shown to be equivalent to the Bragg formulation. The two formulations are compared in Fig. 2B for planes with Miller indices (110). What is important in diffraction is the difference in path length between x-rays scattered from two atoms. The distance si + s2 in the Laue formulation is the same as the distance 2s shown for the Bragg formulation. The Laue approach is by far the more useful one for complicated problems and leads to the concept of the reciprocal lattice (Blaurock, 1982 Warren, 1969) and the reciprocal lattice vector S = Q 14n that makes it possible to create a representation of the crystal lattice in reciprocal space. 

I shall introduce here a concept that is common to all methods. The Brillouin zone (BZ) is described in most solid state physics textbooks. It is defined as a reciprocal lattice cell bounded by the planes that are perpendicular bisectors of the vectors from the origin to the reciprocal lattice points. Fig. 2 illustrates the first BZ cell for a hexagonal lattice. 

The concept of a reciprocal lattice was first introduced by Ewald and it quickly became an important tool in the illustrating and understanding of both the diffraction geometry and relevant mathematical relationships. Let a, b and c be the elementary translations in a three-dimensional lattice (called here a direct lattice), as shown for example in Figure 1.4. A second lattice, reciprocal to the direct lattice, is defined by three elementary translations a"", b and c, which simultaneously satisfy the following two conditions ... 

The Bragg equation shows that diffraction occurs when the scattering vector equals a reciprocal lattice vector. The scattering vector depends on the geometry of the experiment whereas the reciprocal lattice is determined by the orientation and the lattice parameters of the crystalline sample. Ewald s construction combines these two concepts in an intuitive way. A sphere of radius 1//1 is constructed and positioned in such a way that the Bragg equation is satisfied, and diffraction occurs, whenever a reciprocal lattice point coincides with the surface of the sphere (Figure 1.8). 

The concept of anisotropic fluctuations leads to a simple explanation of the epitaxial relationships observed in the transition from one ordered phase to another (Schulz et al., 1994 Koppi et al., 1994). Because of the broken symmetry in a periodically ordered structure, the dominant fluctuation modes reside at specific locations in the reciprocal space, determined by the symmetry of the reciprocal lattice of the initial ordered structure. This spatial relationship is maintained when the new structure—dictated by the most unstable fluctuation modes—grows out of the initial structure, giving rise to the observed epitaxy. 

Many of the physical properties of crystals, as well as the geometry of the three-dimensional patterns of radiation diffracted by crystals, (see Chapter 6) are most easily described by using the reciprocal lattice. The two-dimensional (plane) lattices, sometimes called the direct lattices, are said to occupy real space, and the reciprocal lattice occupies reciprocal space. The concept of the reciprocal lattice is straightforward. (Remember, the reciprocal lattice is simply another lattice.) It is defined in terms of two basis vectors labelled a and b. ... 

Another way to visualise the phenomenon of diffraction is to introduce the concept of the so-called reciprocal lattice. Mathematically, diffraction involves Fourier transformations between the physical arrangement of atoms and the resulting intensities observed in directions determined by Bragg s... 

The Laue equations can be recast by using the concept of the reciprocal lattice. New vectors a, b and c can be defined by the following relationships ... 

In solving three-dimensional triperiodic diffraction problems the concept of the reciprocal lattice (128) helps greatly. Reciprocal space constructions are useful for diperiodic structures also. In the simplest case of a strictly two-dimensional single layer grating, the reciprocal space construction is an array of parallel rods normal to the plane of the grating. These rods cut the plane at the points of a reciprocal net generated by translations of unit reciprocal vectors a and b having properties defined below in terms of the real-space imit vectors d and b (Section IIB). 

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

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